The parity-check matrix \(H\)  is a \((\lceil \frac{n}{r}\rceil +d-1) \times (n+\lceil \frac{n}{r}\rceil )\)  matrix and the rows of \(H\)  are linearly independent from linear algebra. Hence, the dimension of \(\mathcal {C}\)  is \(n-d+1\) .
From Lemma REF  and [1]}, the minimum distance of \(\mathcal {C}\)  is at least \(d+1\) . Finally, the locality of \(\mathcal {C}\)  is \(r\)  from the parity-check matrix \(H\)  and Corollary REF .
